Theorem rexlimddvcbv 44563

Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44561. The equivalent of this theorem without the bound variable change is rexlimddv 3145. Usage of this theorem is discouraged because it depends on ax-13 2377, see rexlimddvcbvw 44562 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rexlimddvcbv.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃)
rexlimddvcbv.2	⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)
rexlimddvcbv.3	⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))

Assertion

Ref	Expression
rexlimddvcbv	⊢ (𝜑 → 𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑦   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)

Proof of Theorem rexlimddvcbv

Step	Hyp	Ref	Expression
1		rexlimddvcbv.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃)
2		rexlimddvcbv.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))
3		rexlimddvcbv.2	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)
4	2, 3	rexlimdvaacbv 44561	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓))
5	1, 4	mpd 15	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063

This theorem is referenced by: (None)